Maximum of an Airy process plus Brownian motion and memory in Kardar-Parisi-Zhang growth.

Abstract

We obtain several exact results for universal distributions involving the maximum of the Airy_{2} process minus a parabola and plus a Brownian motion, with applications to the one-dimensional Kardar-Parisi-Zhang (KPZ) stochastic growth universality class. This allows one to obtain (i) the universal limit, for large time separation, of the two-time height correlation for droplet initial conditions, e.g., C_{∞}=lim_{t_{2}/t_{1}→+∞}h(t_{1})h(t_{2})[over ¯]^{c}/h(t_{1})^{2}[over ¯]^{c}, with C_{∞}≈0.623, as well as conditional moments, which quantify ergodicity breaking in the time evolution; (ii) in the same limit, the distribution of the midpoint position x(t_{1}) of a directed polymer of length t_{2}; and (iii) the height distribution in stationary KPZ with a step. These results are derived from the replica Bethe ansatz for the KPZ continuum equation, with a "decoupling assumption" in the large time limit. They agree and confirm, whenever they can be compared, with (i) our recent tail results for two-time KPZ with the work by de Nardis and Le Doussal [J. Stat. Mech. (2017) 0532121742-546810.1088/1742-5468/aa6bce], checked in experiments with the work by Takeuchi and co-workers [De Nardis et al., Phys. Rev. Lett. 118, 125701 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.125701] and (ii) a recent result of Maes and Thiery [J. Stat. Phys. 168, 937 (2017)JSTPBS0022-471510.1007/s10955-017-1839-2] on midpoint position.